US11733429B2 - Controlled design of localized states in photonic quasicrystals - Google Patents
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- US11733429B2 US11733429B2 US17/049,829 US201917049829A US11733429B2 US 11733429 B2 US11733429 B2 US 11733429B2 US 201917049829 A US201917049829 A US 201917049829A US 11733429 B2 US11733429 B2 US 11733429B2
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- G02B1/00—Optical elements characterised by the material of which they are made; Optical coatings for optical elements
- G02B1/002—Optical elements characterised by the material of which they are made; Optical coatings for optical elements made of materials engineered to provide properties not available in nature, e.g. metamaterials
- G02B1/005—Optical elements characterised by the material of which they are made; Optical coatings for optical elements made of materials engineered to provide properties not available in nature, e.g. metamaterials made of photonic crystals or photonic band gap materials
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- G02B6/00—Light guides; Structural details of arrangements comprising light guides and other optical elements, e.g. couplings
- G02B6/10—Light guides; Structural details of arrangements comprising light guides and other optical elements, e.g. couplings of the optical waveguide type
- G02B6/12—Light guides; Structural details of arrangements comprising light guides and other optical elements, e.g. couplings of the optical waveguide type of the integrated circuit kind
- G02B6/122—Basic optical elements, e.g. light-guiding paths
- G02B6/1225—Basic optical elements, e.g. light-guiding paths comprising photonic band-gap structures or photonic lattices
Definitions
- This invention relates generally to the field of quasicrystalline structures.
- Crystalline materials have long been exploited in many optical and electronic applications for physical properties arising from their crystalline symmetry. Although such crystalline materials allow many technological applications to be fulfilled, there are limitations imposed by such crystalline symmetry. The limited set of distinct symmetries available for crystalline arrangements require a very large contrast in dielectric constant to achieve a full photonic bandgap, and these symmetries result in optical materials whose optical properties are very sensitive to structural and chemical defects.
- the invention relates to a method for generating a (two and) three-dimensional dielectric quasicrystal heterostructure with a photonic bandstructure, comprising a) obtaining quasicrystal tilings, and b) generating a dielectric quasicrystal heterostructure from said quasicrystal tilings, wherein said dielectric quasicrystal heterostructure has a photonic bandstructure that contains degenerate, effectively localized states, lying inside a bandgap.
- the tilings are (two dimensional) pentagonal quasicrystal tilings.
- the tilings can be used to generate two-dimensional heterostructure or each tile can be extruded to form a prism resulting in a three-dimensional heterostructure.
- said pentagonal quasicrystal tilings are obtained as direct projections using a rhombic-icosahedron window from a five-dimensional hypercubic lattice.
- said quasicrystal tilings are obtained as duals to an overlapping set of five periodically spaced grids.
- said dielectric quasicrystal heterostructure of step b) is generated without introducing defects into the heterostructure.
- said degenerate, effectively localized states have precisely predictable and tunable properties.
- said precisely predictable and tunable properties are selected from the group consisting of frequencies, frequency splittings, and spatial configurations.
- the invention relates to a three-dimensional dielectric quasicrystal heterostructure with a photonic bandstructure that contains degenerate, effectively localized states, lying inside a bandgap.
- said quasicrystal heterostructure is defect-free.
- the invention relates to a method for generating two- and three-dimensional dielectric quasicrystal heterostructures with a controlled pre-selected bandgap properties, comprising a) generating a family of distinct defect-free quasicrystal patterns with a pre-selected symmetry; and b) generating a dielectric quasicrystal heterostructures from said quasicrystal patterns wherein the said dielectric quasicrystal heterostructures have photonic band structures photonic band gaps spanning pre-selected frequencies and localized states at pre-selected frequencies inside the bandgap.
- the tilings are (two dimensional) pentagonal quasicrystal tilings.
- the tilings can be used to generate two-dimensional heterostructure or each tile can be extruded to form a prism resulting in a three-dimensional heterostructure.
- said pentagonal quasicrystal tilings are obtained as direct projections using a rhombic-icosahedron window from a five-dimensional hypercubic lattice.
- said quasicrystal tilings are obtained as duals to an overlapping set of five periodically spaced grids.
- said degenerate, effectively localized states have precisely predictable and tunable properties.
- said precisely predictable and tunable properties are selected from the group consisting of frequencies, frequency splittings, and spatial configurations.
- a “photonic quasicrystal” means a quasicrystal that is capable of allowing the transmission, steering, manipulation, and control of some electromagnetic radiation. It is not intended that the term be limited to the transmission of electromagnetic radiation in the visible region. It is also not intended to be capable of transmitting of all electromagnetic radiation.
- the photonic quasicrystal refracts, reflects, defracts, or absorbs electromagnetic radiation at (some) individual frequencies. In preferred embodiments, the photonic quasicrystal refracts, reflects, defracts, or absorbs electromagnetic radiation at pre-selected frequencies.
- a “heterostructure” means a dielectric structure with one or more interface(s) across which the chemical composition changes.
- the interface of the two dielectrics contains a scattering centre in which light propagates more slowly. If the scattering centers are regularly arranged in a medium, light is coherently scattered. In this case, interference causes some frequencies not to be allowed to propagate, giving rise to forbidden and allowed bands. Regions of frequency may appear that are forbidden regardless of the propagation direction.
- “heterostructure” means a structure composed of two or more substances with different dielectric constants. The differences in dielectric constant are due to differences in chemical composition. In some embodiments, one of the substances is air or vacuum. Light propagates at different speeds in media with different dielectric constants.
- an electromagnetic radiation wave scatters from the interfaces between the substances. Interference among the scattered waves causes some frequencies not to be allowed to propagate, giving rise to forbidden and allowed frequency bands. For some frequency bands, propagation is forbidden regardless of the direction or polarization of the incoming electromagnetic wave.
- a “photonic bandgap” material or structure means that for a certain range of wavelengths, no states exist in the structure for electromagnetic radiation to occupy. Electromagnetic radiation with these wavelengths is forbidden in the structure and cannot propagate. The presence of a single point defect, i.e., part of the structure in which the electromagnetic radiation can propagate, generally results in a “localized state”, i.e., a tightly confined region of light energy which must stay within the defect, since it cannot propagate in the structure, and provided the energy is not being absorbed by the material. By introducing defects, one can introduce allowed energy levels in the gap.
- Defects can create waveguides with directional control (e.g., one micron radius, 90 degree bends with 98 percent transmission efficiency), drop/add filters, multiplexors/demultiplexors, resonators, and laser cavities.
- directional control e.g., one micron radius, 90 degree bends with 98 percent transmission efficiency
- drop/add filters e.g., one micron radius, 90 degree bends with 98 percent transmission efficiency
- multiplexors/demultiplexors e.g., 90 degree bends with 98 percent transmission efficiency
- resonators e.g., resonators
- laser cavities e.g., a laser cavities.
- certain defect-free quasicrystal patterns have band gaps that include localized states in the band gap.
- a “quasicrystalline lattice structure” means that the quasicrystal is in the form of material patterned with an open framework.
- the quasicrystal is made by stereolithography in which polymerization produces rhombic or rhombohedral cells characterized by rods that creates an open framework.
- a “dielectric resonator” or “cavity” means a device arranged that allows electromagnetic radiation to propagate back and forth and build up intensity.
- An “optical resonator” or “resonant optical cavity” means an arrangement of optical components which allow a light beam to propagate back and forth and build up intensity. For example, if a mirror is partly transparent one can feed light from outside into the cavity. Two highly reflecting low-less reflectors may be positioned with their reflecting surfaces facing one another to form the cavity. A collimated laser beam enters the cavity and the wavelength of the incident light is rapidly swept in time. At specific wavelengths and at specific feature positions, light resonates within the cavity, building up energy, corresponding to a peak in the transmitted light.
- “Visible spectrum” means electromagnetic radiation that ranges from approximately 780 nanometers (abbreviated nm) to approximately 380 nm.
- nm nanometers
- a regular incandescent bulb produces light within the visible spectrum. It also wastes a lot of its energy radiating invisible radiation, too.
- the photonic quasicrystal can be tailored so that it radiates almost all of its light in the desired visible with little (not really zero) waste. This is the reason for saying “substantially only” with regard to emitting light in the visible region.
- An “interconnect” means a physical attachment between two or more objects.
- a “preselectable rotational symmetry having a characteristic photonic bandgap structure forbidden in a crystalline material” means the structure has at least one five-, seven- or higher-fold symmetry axis and whose bandgap structure exhibits this same symmetry.
- a photonic quasicrystal can have three-dimensional icosahedral symmetry or two dimensional seven-fold symmetry, either of which are impossible for photonic crystals.
- a material having a spherically symmetric property means that rotation by any angle in three dimensions produces no change with regard to the physical property.
- Circularly symmetric means that rotation about an axis by any angle (two dimensions) produces no change.
- Bandgaps are not precisely spherically symmetric (same for circularly symmetric); thus, usage is intended to be substantially so.
- a “quasicrystalline structure includes higher point group symmetry than a crystalline counterpart” means that the quasicrystal can have five-, seven-, eight-, and higher-fold symmetry axes, whereas periodic crystals can never have greater five-fold symmetry or any symmetry greater than six-fold symmetry.
- a crystal, planar hexagonal lattice has six-fold symmetry, the highest and most circular symmetry possible for a crystal or periodic pattern.
- Quasicrystals allow higher, more circular symmetries, such as patterns with seven-, eleven-, forty-seven- or even higher symmetries, Similarly in three-dimensions, the highest symmetry possible for a periodic pattern or crystal is cubic symmetry, whereas quasicrystals can have icosahedral symmetry, which includes five-fold symmetries and which is more spherically symmetric.
- Quasicrystal patterns are classified according to rotational symmetry and “local isomorphism class” or “LI class.”
- Two defect-free quasicrystal patterns in the same LI class are composed of the same units (e.g., tiles) such that every finite arrangement found in one pattern is found with the same frequency in the other and vice versa.
- Two defect-free quasicrystal patterns are in different LI classes if some finite arrangements of units (e.g. tiles) are in one pattern but not in the other, or vice versa.
- Defect-free quasicrystal patterns of a given symmetry and a given set of units, e.g., tile shapes can be constructed in an infinite variety of distinct LI classes.
- a “periodic approximant” is crystalline or periodic pattern whose unit cell or regularly repeating motif consists of a quasicrystal pattern of units e.g., tiles, slightly distorted so it can fit in a periodically repeating array.
- Approximants are a useful technical device for computing the properties of defect-free quasicrystals.
- the photonic properties of a quasicrystal pattern can be computed by first computing the properties for periodic approximants with repeating units containing increasing number of quasicrystal units, e.g., tiles, and determining the limit as the number of units gets large.
- “Achieving higher power efficiency over a selected range of frequencies” means that more of the input power is converted into radiation of the desired frequencies and less is wasted in producing undesired: frequencies.
- an ordinary microwave antenna also broadcasts near infrared and radio waves which are not useful for microwave transmission, but a quasicrystalline antenna would stop the unneeded waves and refocus their power towards producing more microwave radiation at the useful frequencies.
- a “sub-wavelength quasicrystalline structure” means the spacing between the repeating elements in the quasicrystalline structure is smaller than the wavelength of the light.
- a “stealth material” means a radar absorbent material that absorbs the incoming radar radiation without producing any reflections.
- waveguide means any type of transmission line in the sense that it is used to guide electromagnetic radiation from one point to another. Typically, the transmission of electromagnetic energy along a waveguide travels at a velocity somewhat slower than electromagnetic radiation traveling through free space.
- a waveguide may be classified according to its cross section (rectangular, elliptical, or circular), or according to the material used in its construction (metallic or dielectric). Glass fibers, gas-filled pipes, and tubes with focusing lenses are examples of optical waveguides.
- the dielectric constant means the extent to which a substance concentrates the electrostatic lines of flux.
- the dielectric constant also determines the speed of light through the material or, equivalently, the refractive index.
- Substances with a low dielectric constant include a perfect vacuum, dry air, and most pure, dry gases such as helium and nitrogen.
- Materials with moderate dielectric constants include ceramics, distilled water, paper, mica, polyethylene, and glass.
- Metal oxides in general, have high dielectric constants.
- the refractive index is related to the dielectric constant of a material.
- the refractive' index (or index of refraction) of a material is the factor by which the phase velocity of electromagnetic radiation is slowed in that material, relative to its velocity in a vacuum.
- an electromagnetic wave's phase velocity is slowed in a material because the electric field creates a disturbance in the charges of each atom (primarily the electrons) proportional to the permittivity.
- the charges in general, oscillate slightly out of phase with respect to the driving electric field. The charges thus radiate their own electromagnetic wave that is at the same frequency but with a phase delay.
- the macroscopic sum of all such contributions in the material is a wave with the same frequency but shorter wavelength than the original, leading to a slowing of the wave's phase velocity. Most of the radiation from oscillating material charges will modify the incoming wave, changing its velocity.
- FIG. 1 A shows examples of periodic approximants from two different LI classes with pentagonal symmetry (top to bottom) and from three different degrees of approximant (left to right).
- the spectrum of LI classes for these tilings can be labeled by a single number “ ⁇ ” that varies continuously from 0 to 1.
- the unit cell for each approximant is outlined in dashed red lines.
- the tiles that form the unit cell are filled in, with obtuse tiles filled in with green and acute tiles filled in with yellow. (Some of the unit-cell tiles extend beyond the dashed red lines, because we have chosen here to completely fill in tiles (without repeats) that occur at the boundary of the unit cell, instead of truncating them.)
- Dielectric cylinders filled in with green
- FIG. 2 shows the four special vertex environments.
- FIG. 3 shows a representative electric field distribution for the six observed types of effectively localized states. Blue/red/white corresponds to minimum/maximum/zero power for a given state. Contours of dielectric cylinders are shown, and the vertex environments are overlaid (see FIG. 2 ). (a) X, (b) Y, (c) Z 1 , (d) Z 2 , (e) ST 1 , and (f) ST 2 .
- FIG. 4 A show the density F of the four special vertex environments (shown in FIG. 2 ), versus LI class ⁇ .
- FIG. 4 B show Expected fraction ⁇ of special states (shown in FIG. 3 ) versus LI class ⁇ .
- FIG. 5 show Upper band edge frequency ⁇ + ⁇ L N (blue circles, top), lower band edge frequency ⁇ ⁇ and ⁇ N-n loc (red triangles, bottom), and the central frequency (green squares, middle), versus ⁇ .
- ⁇ + and ⁇ correspond to the upper and lower edges, respectively, of the fundamental bandgap.
- Dashed lines represent the aver-age value for a given curve.
- FIG. 6 shows the average frequencies of effectively localized states (shown in FIG. 3 ).
- the shaded, solid regions at the upper and lower ends represent the continuum of states adjacent to the fundamental bandgap.
- the right panel which is a blown up portion of the left panel around the effectively localized states, identifies which effectively localized states correspond to the different frequencies.
- FIG. 7 A shows expected TM spectrum around the fundamental bandgap versus LI class ⁇ . Average frequencies of effectively localized states shown within each range of ⁇ that their corresponding special vertex environments (SVEs) appear.
- SVEs special vertex environments
- FIG. 7 B shows the expected outer bandgap (ratio of frequency gap to the frequency at the midpoint of the gap) versus LI class ⁇ .
- FIG. 8 shows estimating the uncertainty from resolution d ⁇ .
- the bandstructure is computed, first at 512 ⁇ 512 resolution, then at 1024 ⁇ 1024 resolution.
- the relative change in the frequencies ⁇ i (defined in Equation 7) for states in the minibands is shown here (colored circles, left vertical axis). Each position along the horizontal axis corresponds to a fixed rendition from an LI class ⁇ and with dielectric constant ⁇ .
- the LI class is labeled above each subpanel.
- the dielectric constant is shown as a+, with values along the right vertical axis.
- FIG. 9 shows Estimating band width ⁇ .
- the bandstructure is computed.
- the different rows correspond to the different minibands. Each row has been divided into subpanels according to the number of SVE sites n of the miniband type (e.g., for the first row, n is the number of ST 1 sites and, for the second row, n is the number of Z 1 sites). The values of n are shown at the top of each subpanel. Within each subpanel, the renditions are ordered according to increasing ⁇ .
- FIG. 10 shows Electric-field energy density E 2 of an ST 1 MS state versus distance r from central vertex of ST site.
- the distance r is in units of the tile edge length a.
- (Left) The electric field distribution in unit cell for an MS state from the ST 1 miniband, overlaid on the point pattern. Blue/red/white corresponds to negative/positive/zero field, respectively.
- (Upper Right) The maximum value of E 2 at distance r from Site 1 is shown as a red, solid line. This quantity is defined in Equation 11.
- the average value of E 2 at distance r from Site 1 is shown as a black, dotted line. This quantity is defined in Equation 12.
- FIG. 11 shows Electric-field energy density E 2 of an ST 2 SS state versus distance r from central vertex of ST site. Quantities that are described in the FIG. 10 caption are presented here for an ST 2 SS state.
- FIG. 12 shows Electric-field energy density E 2 of an X MS state versus distance r from central vertex of X site. Quantities that are described in the FIG. 10 caption are presented here for an X MS state.
- FIGS. 13 A &B shows SVEs for center-decorated structures ( FIG. 13 A ) and for Delaunay-decorated structures ( FIG. 13 B ). Overlaid on the SVEs are the scatterer configurations.
- FIGS. 14 A &B show representative examples of effectively localized states in center decorated structures.
- FIG. 14 A shows examples of states that have electric field concentrated in the dielectric component (i.e., within scatterers).
- FIG. 14 B shows examples of states that have electric field concentrated in the air component (i.e., between scatterers). Blue/red/white corresponds to negative/positive/zero field, respectively.
- FIG. 15 shows an example of a dielectric structure (right panel) derived from the tiling shown in the left panel.
- the unit cell is outlined in dashed red lines.
- Dielectric cylinders filled in with green
- This invention relates generally to the field of quasicrystalline structures.
- Tilings and definitions The tilings are obtainable as direct projections from a five-dimensional hypercubic lattice or as duals to an overlapping set of five periodically spaced grids [13-15]. We use periodic approximants to compute the bandstructure and verify convergence with the level of approximant.
- FIG. 1 A Examples from different LI classes are shown in FIG. 1 A .
- Penrose LI class this procedure minimizes the density of defects—necessary to make the tilings periodic—to two mismatched edges per unit cell [17].
- n ⁇ the approximants approach the ideal tiling; the number of vertices in the unit cell increases; ⁇ n ⁇ (1+ ⁇ 5)/2 ⁇ 1.618, the golden ratio; and ⁇ i (n) ⁇ r i , the star vectors of the ideal tiling.
- the tilings are composed of two types of rhombuses, both with the same edge length a, but one with interior angle 2 ⁇ /5 (“obtuse”) and the other 2 ⁇ /10 (“acute”).
- obtuse interior angle 2 ⁇ /5
- acute interior angle 2 ⁇ /10
- the ratio of the number of obtuse rhombi to the number of acute rhombi is equal to ⁇ for all tilings; hence, in the n ⁇ limit, all tilings have the same number density, i.e., the same number of vertices per unit area.
- a vertex environment is a configuration of tiles that shares a common vertex.
- the X, Y, Z, and ST vertices (using the notation of Refs. [13, 14, 19]) play an important role in our discussion. They are shown in FIG. 2 , and we refer to them as special vertex environments (SVEs).
- Maxwell's equations are solved for states with transverse magnetic (TM) polarization, i.e., with the electric field oriented parallel to the cylindrical axis (the z-axis in FIG. 1 B ).
- the TM bandstructure contains states in which the electric field is highly concentrated on the SVE, either on one isolated site or on many sites.
- FIG. 3 A-F shows representative examples of these states on isolated sites.
- the number of effectively localized states is directly related to the number of SVEs. We empirically observe that there is one state for every X vertex ( FIG. 3 A ), one for every Y ( FIG. 3 B ); two for every Z ( FIG. 3 C and FIG. 3 D ); and three for every ST ( FIG. 3 E and FIG. 3 F ; there are two orthogonal states that look like FIG. 3 E ).
- the total number of effectively localized states n loc is given by n loc ⁇ N X +N Y +2 N Z +3 N ST , (Equation 3) where N V is the number of SVEs of type V. For different renditions from the same LI class, the number will differ.
- ⁇ i L , ⁇ i H be the lower and upper frequencies of the ith band. It is useful to define, for a given tiling, the upper band edge frequency ⁇ + and the lower band edge frequency ⁇ ⁇ as follows: ⁇ + ⁇ ⁇ L N , ⁇ ⁇ ⁇ H N ⁇ n loc (Equation 5) where N is the number of vertices in the unit cell.
- FIG. 5 shows ⁇ + , ⁇ ⁇ , and their average, plotted for several samples from different LI classes ⁇ , for different degrees of approximants.
- FIG. 6 shows the average midband frequency for each type of state.
- all states of a given type have the same frequency (i.e., are degenerate), and frequencies for different types do not overlap. This indicates that each type has a characteristic frequency.
- Penrose LI class is exceptional for being the only class with no effectively localized TM states; as a consequence, it has the largest outer bandgap. All other LI classes have, generically, effectively localized TM states within the fundamental bandgap with predictable and tunable degeneracies ( FIG. 4 B ) and frequencies ( FIG. 6 ) and are related to the presence of SVEs. Our initial studies using other choices of dielectric decoration show qualitatively the same results, although some choices also produce effectively localized states within the air component of the heterostructure.
- a quasicrystal pattern is restorable if it can be uniquely specified given only a set of rules that fix the allowed clusters within a circle whose radius is smaller than some bound [9].
- the bound can be used to derive a lower limit on the density of configurations of any given size, including the special vertex environments (SVEs).
- the localization sites occur with non-negligible density and, hence, may be multifractal critical states [24, 25] rather than localized in the strict sense.
- ⁇ may be varied such that the densities of some configurations can be made arbitrarily small, as illustrated in FIG. 4 for the case of SVEs. If the configuration is the site of a localized state (say, with field falling away exponentially from the center, as suggested by the simulations for SVEs), then the state is strictly localized in the limit that the density approaches zero.
- Example 2 we examine whether the states within a miniband have a measurable difference in frequency. We identify and estimate the uncertainty stemming from discretization of the unit cell to numerically solve Maxwell's equations for the photonic bandstructure. The band width (i.e., the difference between the maximum and minimum eigenfrequencies in a miniband) is also estimated. We observe that the uncertainty is typically larger than the band width and, therefore, the frequencies of states within the miniband cannot be discriminated from one another (if the differences are theoretically nonzero). The results are also consistent with the states forming a miniband being degenerate in frequency.
- the band width i.e., the difference between the maximum and minimum eigenfrequencies in a miniband
- SS single site
- MS multiple site
- a characteristic feature of a localized state is an exponential falloff of the energy density from the localized site.
- Example 3 we check whether the exponential falloff is observed in examples of SS and MS states to determine to what extent the states are localized. The results show that both SS and MS states are composed of exponentially localized field configurations, which are centered on individual SVE sites.
- the number of pixels N ⁇ N per unit cell which we call the resolution, is a simulation parameter that can be changed.
- N ⁇ limit assuming the pixels are uniformly distributed in the unit cell
- the pixelated unit cell approaches the ideal unit cell.
- N is necessarily finite. Therefore, the discretized unit cell is always an approximation of—and never equal to—the analytically defined unit cell. This approximation leads to some amount of numerical uncertainty in the computed values of frequency ⁇ . We call this the uncertainty from resolution and denote it by d ⁇ .
- Equation ⁇ ⁇ 7 gives an estimate of d ⁇ / ⁇ .
- Each miniband is divided into subpanels according to the number of SVEs n in the rendition of the miniband type (e.g., for the ST 1 miniband shown in the topmost plot, n is the number of ST 1 sites). Within each subpanel, the value of n is shown, and the renditions are ordered according to increasing ⁇ .
- the minibands with the possible exception of the X miniband—have a band width ⁇ that does not systematically increase as the number of SVEs increases. Moreover, it appears that ⁇ ⁇ 10 ⁇ 4 . (Equation 10)(4.15) (The ST2 band is an exception, which has band width 10 ⁇ 5 .)
- a characteristic feature of a localized state is an exponential falloff of the energy density from the localized site.
- we check whether the exponential falloff is observed in examples of SS and MS states to determine to what extent the states are localized. Three states are examined. The states are frequency eigenstates of the same rendition (i.e., the same phases ⁇ i and same approximant) of LI class ⁇ 0:45.
- E 2 (r) be the electric-field energy density of a frequency eigenstate from a miniband, where r denotes the position in the unit cell.
- x denote the position of the central vertex of one of the sites of the SVE type corresponding to the miniband.
- the first state we consider is an ST 1 MS state. Its electric field distribution E(r) is shown in the left panel of FIG. 10 . There are three ST sites in this rendition and the field is non-negligible on each site. For illustrative purposes, we show the quantities defined in Equation 11 and Equation 12 evaluated (right panel of FIG. 10 ) at two of the three ST sites.
- the maximum energy density at distance r, computed according to Equation 11, is shown as red solid lines versus r.
- the average energy density at distance r, computed according to Equation 12 is shown as black, dotted lines. Both quantities have been normalized so that the peak value is set to one.
- the normalized profiles of the curves appear to be virtually identical.
- the log-linear behavior and similarities of the curves for the two ST sites suggest that the energy density is exponentially localized on individual ST sites.
- the background energy density which is computed in the same way as it was for the ST 1 MS state, is ⁇ 10 ⁇ 6 (on the normalized scale).
- the background is ⁇ 10 ⁇ 4 .
- Maxwell's equations are solved for states with transverse magnetic (TM) polarization, i.e., with the electric field oriented parallel to the cylindrical axis (the z-axis in FIG. 15 ).
Abstract
Description
{circumflex over (r)} 0=(1,0), {circumflex over (r)} 1=(cos 2π/5, sin 2π/5),
{circumflex over (r)} 2(n)=(−1,τn −1)·({circumflex over (r)} 0 ,{circumflex over (r)} 1),
{circumflex over (r)} 3(n)=−(τn −1,τn −1)·({circumflex over (r)} 0 ,{circumflex over (r)} 1),
{circumflex over (r)} 4(n)=(τn −1,−1)·({circumflex over (r)} 0 ,{circumflex over (r)} 1),
where τn=Fn+1/Fn (=1/1, 2/1, 3/2, 5/3, . . . ) and Fn is the nth Fibonacci number (F0=F1=1). Examples from different LI classes are shown in
|−½+{γ}|=|−½+{γ′}|, (Equation 1)
where {γ} denotes the fractional part of γ. The distinct values of γ lie within the interval [0, 0.5]; γ=0 corresponds to the Penrose tiling. Any γ can be mapped to an equivalent one γ′ within the interval [0, 0.5]] via
γ′=½−|−½+{γ}|. (Equation 2)
Moreover, γ, γ′ε[0, 0.5] and γ′≠γ′, then γ is not locally isomorphic to γ′.
n loc ≡N X +N Y+2N Z+3N ST, (Equation 3)
where NV is the number of SVEs of type V. For different renditions from the same LI class, the number will differ.
φ(γ)=F X(γ)+F Y(γ)+2F Z(γ)+3F ST(γ), (Equation 4)
where FV is the density of SVEs of type V, shown in
ω+≡ωL N,ω−≡ωH N −n loc (Equation 5)
where N is the number of vertices in the unit cell.
Δω>dω (Equation 6)
gives an estimate of dω/ω.
dω/ω˜3×10−3 (Equation 8)
dω=(dω/ω)ω˜(3×10−3)0.3˜10−3. (Equation 9)
Δω˜10−4. (Equation 10)(4.15)
(The ST2 band is an exception, which has
as well as the average value of the energy density around the circle of radius r centered at x:
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